Masur's Divergence for Tori and Kummer Surfaces

TL;DR

This paper extends Masur's divergence theorem from translation surfaces to complex tori and Kummer surfaces, linking the recurrence of geodesic flow to unique ergodicity of horizontal foliations in these complex geometries.

Contribution

It generalizes Masur's divergence to Kähler moduli spaces of tori and Kummer surfaces, introducing algebraic conditions for ergodicity.

Findings

Horizontal foliation is uniquely ergodic if the geodesic flow is recurrent.

Derived algebraic conditions necessary for ergodicity.

Extended Masur's divergence theorem to complex surfaces.

Abstract

Masur's divergence states that the horizontal foliation of translation surfaces is uniquely ergodic if the geodesic flow is recurrent on the moduli space. This established a relationship between geometrical properties of foliations and the dynamics on the moduli space. In this paper, we extend this theorem to complex torus and Kummer surfaces. We define and calculate horizontal foliations and the corresponding geodesic flow in the moduli space of K\"ahler metrics and prove that the horizontal foliation is uniquely ergodic if the geodesic flow is recurrent. We also find the a necessary algebraic condition on the geodesic flow for the horizontal foliation to be uniquely ergodic.